In this work, we report on the fabrication of continuous transparent and flexible supercapacitors by depositing a CNT network onto a polymer electrolyte membrane directly from an aerogel of ultra-long CNTs produced floating in the gas phase. The supercapacitors combine record power density of $1370 kW kg^{-1}$ at high transmittance ($ca. 70%$), high electrochemical stability during extended cycling ($>94%$ capacitance retention over $20 000 cycles$) as well as against repeated $180{deg}$ flexural deformation. They represent a significant enhancement of 1-3 orders of magnitude compared to the prior-art transparent supercapacitors based on graphene, CNTs, and rGO. These features mainly arise from the exceptionally long length of the CNTs, which makes the material behave as a bulk conductor instead of an aspect ratio-limited percolating network, even for electrodes with $>90%$ transparency. The electrical and capacitive figures-of-merit for the transparent conductor are $FoMe = 2.7$, and $FoMc = 0.46 F S^{-1} cm^{-2}$ respectively
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